Value Adjustment Research Articles

Wrong Way Risk Modeling and Computation in Credit Valuation Adjustment for European and Bermudan Options

We study the impact of wrong-way-risk (WWR) on credit valuation adjustment (CVA) for European and Bermudan options, based on an intensity model. WWR is modeled by a dependency between the underlying asset and the intensity of the counterparty’s default. We consider three different models. We take the difference between the default-free value and the default-adjusted value for the purpose of the CVA calculations. Two numerical algorithms, the COS method and the Stochastic Grid Bundling Method (SGBM) are generalized and employed for the computations. By varying correlation coefficients, we perform a CVA stress test for European options, show differences in the optimal exercise boundaries, in corresponding Bermudan option values and compute the CVA shortfall.

The Basis Goes Stochastic: A Jump-Diffusion Model for Financial Risk Applications

There are several pricing and risk model applications where the assumption of a deterministic LIBOR-OIS basis can lead to severe mispricing. By modeling such a basis using a jump-diffusion process, we show how stochastic basis can impact the valuation of specific deals such as zero-coupon swaps, as well as credit valuation adjustments and gap risk in particular. Explicit formulas are provided and numerical examples showcased.

Arbitrage-Free Pricing of XVA Part II: PDE Representation and Numerical Analysis

We study the semilinear partial differential equation (PDE) associated with the non-linear BSDE characterizing buyer’s and seller’s XVA in a framework that allows for asymmetries in funding, repo and collateral rates, as well as for early contract termination due to counterparty credit risk. We show the existence of a unique classical solution to the PDE by first proving the existence and uniqueness of a viscosity solution and then its regularity. We use the uniqueness result to conduct a thorough numerical study illustrating how funding costs, repo rates, and counterparty credit risk contribute to determine the total valuation adjustment.

Efficient estimation of sensitivities for counterparty credit risk with the finite difference Monte Carlo method

According to Basel III, financial institutions have to charge a credit valuation adjustment (CVA) to account for a possible counterparty default. Calculating this measure and its sensitivities is one of the biggest challenges in risk management. Here, we introduce an efficient method for the estimation of CVA and its sensitivities for a portfolio of financial derivatives. We use the finite difference Monte Carlo (FDMC) method to measure exposure profiles and consider the computationally challenging case of foreign exchange barrier options in the context of the Black-Scholes as well as the Heston stochastic volatility model, with and without stochastic domestic interest rate, for a wide range of parameters. In the case of a fixed domestic interest rate, our results show that FDMC is an accurate method compared with the semi analytic COS method and, advantageously, can compute multiple options on one grid. In the more general case of a stochastic domestic interest rate, we show that we can accurately compute exposures of discontinuous one-touch options by using a linear interpolation technique as well as sensitivities with respect to initial interest rate and variance. This paves the way for real portfolio level risk analysis.

Practical Applications of A Conversation with Danske Bank’s Jesper Andreasen

The world of quantitative finance is facing new demands on its tools, technologies and talents. Driven by regulatory requirements, banks must take capital requirements and valuation adjustments seriously, but the solutions to many of the new mandates are time consuming and labor intensive. Quantitative talent and technology can help, argues Jesper Andreasen of Danske Bank . In a conversation with Institutional Investor Journals , Andreasen offers his views on the challenges and opportunities created by an increasingly compliance-driven environment. He is speaking on the subject of capital valuation adjustments (KVA) and time series estimation of stochastic models at Global Derivatives .

Option-Based Pricing of Wrong Way Risk for CVA

The two main issues for managing wrong way risk (WWR) for the credit valuation adjustment (CVA, i.e. WW-CVA) are calibration and hedging. Hence we start from a novel model-free worst-case approach based on static hedging of counterparty exposure with liquid options. We say "start from" because we demonstrate that a naive worst-case approach contains hidden unrealistic assumptions on the variance of the hazard rate (i.e. that it is infinite). We correct this by making it an explicit (finite) parameter and present an efficient method for solving the parametrized model optimizing the hedges. We also prove that WW-CVA is theoretically, but not practically, unbounded. The option-based hedges serve to significantly reduce (typically halve) practical WW-CVA. Thus we propose a realistic and practical option-based worst case CVA.

Uncollateralised Interest Rate Swap xVA

Derivative valuation adjustments (xVA) have been an evolving topic over the past decade, now giving us a rich theory of what was traditionally referred to as the relatively opaque "credit spread" imposed by banks. As independent risk advisors, we are regularly positioned between clients and their banks, considering the impacts of both parties' xVA when entering into and terminating positions. This paper documents our implementation of xVA modelling at the time of writing, specifically for the derivatives which our clients trade the most: uncollateralised interest rate swaps. It is also one of the first open demonstrations of an independent party modelling banks' xVA at levels actually observed by them. We focus on the most significant credit (CVA), funding (FVA) and capital (KVA) valuation adjustments. Our full modelling framework and its calibration are described and examples are given for swaps under the Standardised Approach (SA-CCR) capital framework, effective 1 January 2017.

Capturing Initial Margin in Counterparty Risk Calculations

This paper compares a range of alternative approaches to incorporate Initial Margins (IMs) in the modelling of counterparty credit risk exposures. With the rise of Central Counterparties to clear OTC derivatives and the incoming legislation requiring bilateral margining for uncleared derivatives between financial counterparties the reflection of IMs becomes an essential model component that drives exposures, associated regulatory capital requirements and valuation adjustments such as CVA and MVA. The influence of the modelling choices is explored by means of typical derivatives portfolios. For the actual estimation of a path-dependent (“stochastic”) IM through time the use of quantile regression is suggested as an econometrically reliable approximation. Banks’ internal counterparty risk models will likely exhibit a basis vis-a-vis the actual IM mechanisms in practice (for example, owing to different risk factor representations and/or calibrations). In this context, the paper suggests that a simplified representation in the form of a “dynamic IM” can approximate most of the quantities of interest to a reasonable degree.

Prudent Valuation Guidelines and Sound Practices

The EU Capital Requirement Regulation (CRR) and the EBA Regulatory Technical Standard for prudent valuation, published on Jan. 1st, 2014 and Jan. 28th, 2016, respectively, constitute the EU Prudent Valuation Framework, which represents a challenge for financial institutions both from a methodological and practical point of view. In this work we seek an ambitious fourfold result. First, we work out a practical realization of the regulatory requests, smoothing the difficulties arising from the detailed interpretation of the rules. Second, we give a comprehensive methodological analysis of the different sources of valuation risk (ambiguity, liquidity, correlation, concentration and model risks) that bring about the need of computing valuation adjustments. Third, we explore a wide range of possible individual AVA calculation approaches, providing many practical case studies mimicking the actual valuation scenarios that institutions might face. Finally, we design a comprehensive example of the entire Prudent Valuation process, covering both organizational and IT issues. This work provides a set of guidelines and sound practices that could be used as a starting point and, possibly, a benchmark for prudent valuation. On the one side, institutions will benefit of the rules explained and applied into a variety of case studies. On the other side, regulators will benefit to see how the rules are interpreted and applied by institutions. We do not aim either to give any authentic interpretation of the regulations, which ultimately belongs to its owners, or to question the principles behind the regulations, since real challenges may come out only from practical applications of the principles to real financial portfolios, and comparisons across different financial institutions.This work is the result of the joint effort of the Market Risk Committee of AIFIRM - Italian Association of Financial Industry Risk Managers.

A bond consistent derivative fair value

In this paper we present a rigorously motivated pricing equation for derivatives, including general cash collateralization schemes, which is consistent with quoted market bond prices. Traditionally, there have been differences in how instruments with similar cash flow structures have been priced if their definition falls under that of a financial derivative versus if they correspond to bonds, leading to possibilities such as funding through derivatives transactions. Furthermore, the problem has not been solved with the recent introduction of Funding Valuation Adjustments in derivatives pricing, and in some cases has even been made worse. In contrast, our proposed equation is not only consistent with fixed income assets and liabilities, but is also symmetric, implying a well-defined exit price, independent of the entity performing the valuation. Also, we provide some practical proxies, such as first-order approximations or basing calculations of CVA and DVA on bond curves, rather than Credit Default Swaps.

FVA and CVA for Collateralized Trades with Re-hypothecation

Through this article we join the ongoing “FVA debate” with a strategy of bilateral replication, which enables us to price derivatives trades and identify XVAs, including the FVA. We let the issuer and buyer replicate the derivative's payoffs and the loss given the issuer's default, respectively, while taking into account the costs of funding for replication and collateral posting in the replicating portfolios. The strategy leads to a PDE for the economic value of the derivative, and the PDE also implies the unique risk-neutral pricing measure. A proper solution of the PDE gives rise to a decomposition of the derivatives economic value into the risk-free value of the derivative, the credit valuation adjustment, and the funding valuation adjustment. We show that in general the economic values to the two trading parties are different, due to the different funding needs and funding costs, and both parties may have to bear some funding costs in order to strike a deal. When the premium for buying a derivatives asset is also funded, then to the buyer the trade actually consists of two transactions: borrowing and purchasing, and the FVA for the trade can be obtained by separately evaluating the FVAs of the two transactions.

Multi-Curve Discounting

We introduce the multi-curve discounting (MCD) method, where the discount curve depends on the liquidity horizon of the asset. The difference between the value of an asset using OIS discounting and a discount curve referencing the liquidity horizon can be interpreted as a Funding Valuation Adjustment (FVA). We show that a simple model for liquidity risk implies MCD. The liquidity risk model formulation clarifies how a non-zero FVA occurs without violating the Modigliani-Miller theorem.

FIXING RISK NEUTRAL RISK MEASURES

In line with regulations and common risk management practice, the credit risk of a portfolio is managed via its potential future exposures (PFEs), expected exposures (EEs), and related measures, the expected positive exposure (EPE), effective expected exposure (EEE), and the effective expected positive exposure (EEPE). Notably, firms use these exposures to set economic and regulatory capital levels. Their values have a big impact on the capital that firms need to hold to manage their risks. Due to the growth of credit valuation adjustment (CVA) computations, and the similarity of CVA computations to exposure computations, firms find it expedient to compute these exposures under the risk neutral measure. Here, we show that exposures computed under the risk neutral measure are essentially arbitrary. They depend on the choice of numéraire, and can be manipulated by choosing a different numéraire. The numéraire can even be chosen in such a way as to pass backtests. Even when restricting attention to commonly used numéraires, exposures can vary by a factor of two or more. As such, it is critical that these calculations be carried out under the real world measure, not the risk neutral measure. To help rectify the situation, we show how to exploit measure changes to efficiently compute real world exposures in a risk neutral framework, even when there is no change of measure from the risk neutral measure to the real world measure. We also develop a canonical risk neutral measure that can be used as an alternative approach to risk calculations.

Gap Risk KVA and Repo Pricing: An Economic Capital Approach in the Black-Scholes-Merton Framework

Although not a formal pricing consideration, gap risk or hedging errors are the norm of derivatives businesses. Starting with the gap risk during a margin period of risk of a repurchase agreement (repo), this article extends the Black-Scholes-Merton option pricing framework by introducing a reserve capital approach to the hedging error's irreducible variability. An extended partial differential equation is derived with two new terms for expected gap loss and economic capital charge, leading to the gap risk economic value adjustment and capital valuation adjustment (KVA) respectively. Practical repo pricing formulae is obtained showing that the break-even repo rate decomposes into cost of fund and economic capital charge in KVA. At zero haircut, a one-year term repo on main equities could command a capital charge as large as 50 basis points for a 'BBB' rated borrower.

Pricing CDS spreads with Credit Valuation Adjustment using a mixture copula

Credit derivatives pricing models before Basel III ignored losses in market value stemming from higher probability of counterparty default. We propose a general credit derivatives pricing model to evaluate a Credit Default Swap (CDS) with counterparty risk, including the Credit Valuation Adjustment (CVA) in order to optimize the economic capital allocation. We work from the model proposed by Luciano (2003, Working Paper, International Center of Economic Research) and the general pricing representation established by Sorensen and Bollier (Financial Analysts Journal 1994;50(3):23–33) to provide a model close to the market practice, easy to implement and fitting with Basel III framework. We approach the dependence between counterparty risk and that of the reference entity with a technical tool: the copula, in particular, the mixture one that combines common “extreme” copulas. We study the CDS's vulnerability in extreme dependence cases. By varying Spearman's rho, the mixture copula covers a broad spectrum of dependence and ensures closed form prices. We end up with an application on real market data.

Conic CVA and DVA

Conic pricing (or bid and ask pricing) of credit risk shows how counterparty credit risk when conservatively valued at the bid price results in larger CVA than would occur under risk neutral pricing. On the other hand when it comes to the debt valuation adjustment, since it is a liability, it must be priced at the ask. This mitigates and can even eliminate the DVA.

Collateral and PDE

After Lehman default (credit crisis 2007), practitioners considered the default risk as a major risk. The regulators pushed the industry to use collateral in order to reduce the risk. In this new world, we want to see how this new considerations affect the theory related to the Partial Differential Equation (PDE) pricing methodology. First, we consider a theoretical framework introduced by Piterbarg, then we relax the perfect collateralisation assumption and establish the correct derivation of the hedgingportfolio.We do not intent to offense anyone and recognise fully the work done in the building of the Post Lehman Theory.We establish different PDE forms dependent of the treasury management strategy and also retrieve Kamtchueng results for the Funding Valuation Adjustment. Indeed there are differences between Funding charge coming from the cash flows due and the funding cost strategy resulting of cash and trading desks.

Capital Valuation Adjustment and Funding Valuation Adjustment

Capital Valuation Adjustment and Funding Valuation Adjustment

Efficient computation of exposure profiles on real-world and risk-neutral scenarios for Bermudan swaptions

This paper presents a computationally efficient technique for the computation of exposure distributions at any future time under the risk-neutral and some observed real-world probability measures; these are needed for the computation of credit valuation adjustment (CVA) and potential future exposure (PFE). In particular,we present a valuation framework for Bermudan swaptions. The essential idea is to approximate the required value function via a set of risk-neutral scenarios and use this approximated value function on the set of observed real-world scenarios. This technique significantly improves the computational efficiency by avoiding nested Monte Carlo simulation and using only basic methods such as regression. We demonstrate the benefits of this technique by computing exposure distributions for Bermudan swaptions under the Hull-White and G2++ models. (Less)

Efficient Computation of Exposure Profiles on Real-World and Risk-Neutral Scenarios for Bermudan Swaptions

This paper presents a computationally efficient technique for the computation of exposure distributions at any future time under the risk-neutral and some observed real-world probability measures; these are needed for the computation of credit valuation adjustment (CVA) and potential future exposure (PFE). In particular, we present a valuation framework for Bermudan swaptions. The essential idea is to approximate the required value function via a set of risk-neutral scenarios and use this approximated value function on the set of observed real-world scenarios. This technique significantly improves the computational efficiency by avoiding nested Monte Carlo simulation and using only basic methods such as regression. We demonstrate the benefits of this technique by computing exposure distributions for Bermudan swaptions under the Hull-White and G2 models.