Value Adjustment Research Articles

Funding Adjustments in Equity Linear Products

Valuation adjustments are nowadays a common practice to include credit and liquidity effects in option pricing. Funding costs arising from collateral procedures, hedging strategies and taxes are added to option prices to take into account the production cost of financial contracts so that a profitability analysis can be reliably assessed. In particular, when dealing with linear products, we need a precise evaluation of such contributions since bid-ask spreads may be very tight. In this paper we start from a general pricing framework inclusive of valuation adjustments to derive simple evaluation formulae for the relevant case of total return equity swaps when stock lending and borrowing is adopted as hedging strategy.

CCR KVA Relief Through CVA: A Regression-Based Monte Carlo Approach

We present and examine, by example of a USD interest rate swap and a EUR/USD cross-currency basis swap, a regression-based Monte Carlo approach to counterparty credit default risk (CCR) capital and CCR capital valuation adjustment (KVA) calculations [assuming the standardized approach to counterparty credit risk for exposure-at-default (SA-CCR EAD) and the internal ratings-based (IRB) approach for CCR risk weights]. This approach allows to incorporate the capital lowering effect of credit valuation adjustment (CVA) in an efficient manner, without having to resort to lengthy nested Monte Carlo simulations. We find that the regression-based Monte Carlo approach works well in most situations. In other situations, the accuracy of the approach is sensitively controlled by the choice of explanatory variables. We discuss in detail the conditions and underlying dynamics under which this happens. In computing and presenting a selection of numerical examples, we also explore the impact of dynamic CCR risk weights on CCR KVA, and compare regression-based CCR KVA results with CCR KVA results from nested Monte Carlo, alternative frequently used CCR KVA simplifications, and standardized CVA KVA.

Two-Factor Hull-White Model Revisited: Correlation Structure for Two-Factor Interest Rate Model in CVA Calculation

The development of credit valuation adjustment (CVA) (valuation adjustments [XVA]) [Green] has increased the importance of simple interest rate models such as the Hull-White model [Tan14] [Tsuchiya]. This is because the XVA model is an FX hybrid model, and is tractable only when the interest rate part is a simple Gaussian model. For the XVA calculation of interest rate instruments, de-correlation of the yield curve can be important even for the swap portfolio. Capturing the correlation structure in the two-factor Hull-White model is an integral element of CVA (XVA) modeling. However, the correlation structure in two-factor Hull-White model has not studied enough except for the analysis in [AndersenPiterbarg]. In this study, the correlation structure of the two-factor Hull-White model is analyzed in detail. The correlation structure of co-initial swap rates is investigated using a combination of the approximation formula and Monte-Carlo simulation. The Hull-White model captures the de-correlation of the yield curve only when the parameters (volatilities and mean reversion strength) satisfy certain relationships, making the valuation of XVA by two-factor Hull-White model effective.

Stochastic approximation schemes for economic capital and risk margin computations

We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja [17] or by regression à la Broadie, Du, and Moallemi [10]. Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in [17] and [10]. Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking parlance. This is illustrated numerically by a KVA case study implemented on GPUs.

Fast stochastic forward sensitivities in Monte Carlo simulations using stochastic automatic differentiation (with applications to initial margin valuation adjustments)

Fast stochastic forward sensitivities in Monte Carlo simulations using stochastic automatic differentiation (with applications to initial margin valuation adjustments)

Counterparty risk: credit valuation adjustment variability and value-at-risk

The third installment of the Basel Accords advocates a capital charge against credit valuation adjustment (CVA) variability. We propose an efficient numerical approach that allows us to compute risk measures for the CVA process by assessing the distribution of the CVA at a given horizon. This approach relies on a recursive formulation of the CVA, yielding the adjustment as a function of both the time to maturity and the value of the risk factors. Numerical experiments are presented to illustrate the impact of various parameters and assumptions on the CVA distribution. More specifically, we investigate the impact of the constant exposure approximation and show that this assumption significantly affects the tail of the distribution of CVA movements. We also find that distortions between physical and risk-neutral probability measures have practically no impact on the dispersion of the CVA distribution. Finally, we analyze the effect of wrong-way risk and of early exercise opportunities on the evaluation of risk measures.

Affine Term-Structure Models: A Time-Changed Approach with Perfect Fit to Market Curves

We address the so-called calibration problem which consists of fitting in a tractable way a given model to a specified term structure like, e.g., yield, prepayment or default probability curves. Time-homogeneous jump-diffusions like Vasicek or Cox-Ingersoll-Ross (possibly coupled with compound Poisson jumps, JCIR), are tractable processes but have limited flexibility; they fail to replicate actual market curves. The deterministic shift extension of the latter (Hull-White or JCIR++) is a simple but yet efficient solution that is widely used by both academics and practitioners. However, the shift approach is often not appropriate when positivity is required, which is a common constraint when dealing with credit spreads or default intensities. In this paper, we tackle this problem by adopting a time change approach. On the top of providing an elegant solution to the calibration problem under positivity constraint, our model features additional interesting properties in terms of implied volatilities. It is compared to the shift extension on various credit risk applications such as credit default swap, credit default swaption and credit valuation adjustment under wrong-way risk. The time change approach is able to generate much larger volatility and covariance effects under the positivity constraint. Our model offers an appealing alternative to the shift in such cases.

An Elementary Approach to the Fundamental Theorems of Asset Pricing in Jump-Diffusion Markets with Credit Risks, Funding Risks, and Collateral

We present an elementary approach to fundamental theorems of asset and derivative pricing for a wide class of jump-diffusion markets in which there are credit risks, funding risks, and collateral. The approach yields new and intuitive proofs of both the first and second Fundamental Theorems of Asset Pricing. We give examples of how the framework incorporates the gamut of industry pricing techniques from simple Black-Scholes pricing to recent results known as Funding Valuation Adjustments (FVA). The main contribution is the insight and intuition into the nature of the fundamental theorems that is afforded by the simplicity of the setup and the consequent reliance on only the most basic results from linear algebra.

Bilateral Defaultable Financial Derivatives Pricing and Credit Valuation Adjustment

The one-side defaultable financial derivatives valuation problems have been studied extensively, but the valuation of bilateral derivatives with asymmetric credit qualities is still lacking convincing mechanism. This paper presents an analytical model for valuing derivatives subject to default by both counterparties. The default-free interest rates are modeled by the Market Models, while the default time is modeled by the reduced-form model as the first jump of a time-inhomogeneous Poisson process. All quantities modeled are market-observable. The closed-form solution gives us a better understanding of the impact of the credit asymmetry on swap value, credit value adjustment, swap rate and swap spread.

AN EQUILIBRIUM MODEL FOR AN OTC DERIVATIVE MARKET UNDER A COUNTERPARTY RISK CONSTRAINT

In this study, we develop an equilibrium pricing model for an option contract with a counterparty risk, a collateral agreement, a counterparty risk constraint, and a threshold. Since we consider the option market to be an example of the derivatives market, we suppose that the buyer of an option has only countertparty risk of a seller defaulting. In addition, we consider a model where the buyer is allowed to enter into an option contract within an allocated amount of risk capital for counterparty risk, and requires (cash) collateral to the seller if the exposure exceeds the threshold. The counterparty risk is measured as a credit valuation adjustment. We provide an equilibrium pricing rule and an equilibrium volume formula by solving participants’ static utility-maximization problems. Based on numerical simulations, we verify the mechanisms through which collateralization, risk capital, and the threshold affect the size of the over-the-counter (OTC) option market. Finally, we analyze the influence of the buyer’s risk-aversion on the market, without collateralization. The results imply that the risk constraint might be a proxy for an investor’s attitude towards risk.

Arbitrage Opportunities in CDS Term Structure: Theory and Implications for OTC Derivatives

Absence-of-Arbitrage (AoA) is the basic assumption underpinning derivatives pricing theory. As part of the OTC derivatives market, the CDS market not only provides a vehicle for participants to hedge and speculate on the default risks of corporate and sovereign entities, it also reveals important market-implied default-risk information concerning the counterparties with which financial institutions trade, and for which these financial institutions have to calculate various valuation adjustments (collectively referred to as XVA) as part of their pricing and risk management of OTC derivatives, to account for counterparty default risks. In this study, we derive No-arbitrage conditions for CDS term structures, first in a positive interest rate environment and then in an arbitrary one. Using an extensive CDS dataset which covers the 2007-09 financial crisis, we present a catalogue of 2,416 pairs of anomalous CDS contracts which violate the above conditions. Finally, we show in an example that such anomalies in the CDS term structure can lead to persistent arbitrage profits and to nonsensical default probabilities. The paper is a first systematic study on CDS-term-structure arbitrage providing model-free AoA conditions supported by ample empirical evidence. Thanks for the feedback from participants at the conferences where the paper was invited for presentation: Machine Learning in Quantitative Finance in Nov. 2018, London, UK.

A SUBORDINATED CIR INTENSITY MODEL WITH APPLICATION TO WRONG-WAY RISK CVA

Credit valuation adjustment (CVA) pricing models need to be both flexible and tractable. The survival probability has to be known in closed form (for calibration purposes), the model should be able to fit any valid credit default swap (CDS) curve, should lead to large volatilities (in line with CDS options) and finally should be able to feature significant wrong-way risk (WWR) impact. The Cox–Ingersoll–Ross (CIR) model combined with independent positive jumps and deterministic shift (JCIR[Formula: see text]) is a very good candidate : the variance (and thus covariance with exposure, i.e. WWR) can be increased with the jumps, whereas the calibration constraint is achieved via the shift. In practice however, there is a strong limit on the model parameters that can be chosen, and thus on the resulting WWR impact. This is because only non-negative shifts are allowed for consistency reasons, whereas the upwards jumps of the JCIR[Formula: see text] need to be compensated by a downward shift. To limit this problem, we consider the two-side jump model recently introduced by Mendoza-Arriaga and Linetsky, built by time-changing CIR intensities. In a multivariate setup like CVA, time-changing the intensity partly kills the potential correlation with the exposure process and destroys WWR impact. Moreover, it can introduce a forward looking effect that can lead to arbitrage opportunities. In this paper, we use the time-changed CIR process in a way that the above issues are avoided. We show that the resulting process allows to introduce a large WWR effect compared to the JCIR[Formula: see text] model. The computation cost of the resulting Monte Carlo framework is reduced by using an adaptive control variate procedure.

Nonlinear valuation under credit, funding, and margins: Existence, uniqueness, invariance, and disentanglement

Since the 2008 global financial crisis, the banking industry has been using valuation adjustments to account for default risk and funding costs. These adjustments are computed separately and added together by practitioners as if the valuation equations were linear. This assumption is too strong and does not allow to model market features such as different borrowing and lending rates and replacement default closeout. Hence we argue that the full valuation equations are nonlinear, and this paper is devoted to studying the nonlinear valuation equations introduced in Pallavicini et al (2011).We illustrate all the cash flows exchanged by the parties involved in a derivative contract, in presence of default risk, collateralisation with re-hypothecation and funding costs. Then we show how to obtain semi-linear PDEs or Forward Backward Stochastic Differential Equations (FBSDEs) from present-valuing said cash flows in an arbitrage-free setup, and we study the well-posedness of these PDEs and FBSDEs in a viscosity and classical sense.Moreover, from a financial perspective, we discuss cases where classical valuation adjustments (XVA) can be disentangled. We show how funding costs are offset by treasury valuation adjustments when one takes a whole-bank perspective in the valuation, while the same costs are not offset by such adjustments when taking a shareholder perspective. We show that although we use a risk-neutral valuation framework based on a locally risk-free bank account, our final valuation equations do not depend on the risk-free rate. Finally, we show how to consistently derive a netting set valuation from a portfolio level one.

The Missing Term in the FVA Calculation

In the aftermath of the Global Financial Crisis, many large banks moved to include Funding Value Adjustment (FVA) in their derivative valuations, despite controversy in the industry over its appropriateness. This paper examines the mathematical derivations of FVA-inclusive valuation formulas in an influential paper advocating FVA inclusion and identifies an implicit assumption that is unrealistic yet crucial, and hence creates problems for the derivation. We examine what happens when the assumption is abandoned and find that it can still be theoretically sound to include FVA in valuations, and to use formulas derived in the paper, subject to some modifications in either the discount rates used, or in what other valuation adjustments are included. Different types of FVA calculation are appropriate when the valuation is from a shareholder-only perspective from when it is from a whole-of-bank perspective (ie including the interests of debtholders). It is also observed that a bank that uses what are here termed 'full-FVA-inclusive valuations', may find that, in cases where the FVA is an asset, prudential supervisors require them to deduct that asset from measures of prudential capital such as CET1.

An Integrated OTC Derivatives Risk Capital Framework

Bilateral derivatives valuation is subject to counterparty credit risk (CCR) in that a counterparty could jump to default or its credit spread could vary over time. In the nomenclature of risk management, the former is called CCR exposure and the later leads to credit valuation adjustment (CVA). As both depend on derivatives valuation, their proper inclusion in risk management has been challenging and controversial. The CVA capital charge, for example, has gone through a few formal implementations since its post crisis inception, and is still subject to wide criticism. Furthermore, the reformed regulatory risk capital framework does not address post-crisis advances in derivatives valuation adjustments (xva). This article presents a holistic OTC derivatives risk capital framework based on an integrated pricing model of market risk and counterparty credit risk. The new framework pulls closer to real economic risk, reduces redundant capital requirements, and cuts implementation costs.

Accelerating CVA and CVA Sensitivities Using Quasi Monte Carlo Methods

We analyze the efficiency of the quasi Monte Carlo method (QMC) when used to compute credit valuation adjustment (CVA) and CVA sensitivities for various portfolios of interest rate swaps using a multi-currency extension to the Hull-White model. We find that QMC with Sobol’ sequences and the Brownian bridge discretization produces results as accurate as classical MC with 10,000 simulations when using roughly 800 simulations: a factor of 12 acceleration. The acceleration varies significantly across portfolios (increasing with moneyness and usually, but not always, decreasing with number of factors), calculation types (order from highest to lowest, usually, but not always, CVA, CR Deltas, IR and FX Deltas, and IR and FX Vegas), and the choice of model (local models usually outperform global models). QMC without the Brownian bridge discretization results in a more modest factor of 4 acceleration. Classical MC with antithetic sampling results in a factor of 2 acceleration. Randomized QMC (linear permutation of digits) with the Brownian bridge discretization improved the efficiency over the non-randomized sequences for smaller in-the-money portfolios. Randomization in combination with a direct and independent simulation approach of the risk factors produced extraordinary results in the limited tests we performed (CVA on a single currency swap), offering an additional factor of 6 acceleration over the pathwise simulation.

The Impact of Size and Leverage on Equity Volatility

Many valuation practitioners rely on equity volatilities when performing equity allocations, derivative valuations, discounts for lack of marketability, or contingent consideration analyses. Yet in contrast to other commonly accepted valuation adjustments,1 practitioners still appear hesitant to adjust historical or implied volatilities from publicly traded comparables for differences in size when performing these types of analyses, despite theory that these factors should impact equity volatility.2 Instead, preference is usually made in the industry to adjust equity volatilities for leverage alone. In this paper I provide analytical support across a twenty-year time horizon using a five-year lookback period to examine the empirical impact of size and leverage on equity volatilities. I find that size has substantial impact on equity volatilities and provide suggested volatility size adjustments for equity volatility. Further, I find evidence that while leverage does have an impact on equity volatilities, ...

Information Technology Value Engineering Model and Cost Efficiency in IT-Based Firms

This paper aims to explore the improved business performance established on optimum costs. This structure rests on a proposed information technology (IT) value engineering model, which is an engineering process associated with IT inclusion in a business organization. Therefore, the methodology approach utilizes systems engineering to develop an engineering scheme procedure by stating the problem, creating alternate resolutions, deciding a resolution, specifying the scheme, and confirming the model. Additionally, this methodology works in cooperation with the resource-based view (RBV) and the partial adjustment valuation (PAV) theories. The RBV recommends that a firm worthy performance can rely on its own resources, whereas PAV reconstructs the input–output relationship of the system. The input consists of K = the regular capital, L = the labor expense, and I = the IT capital, whereas the output is y as the real output of a production process unit. Furthermore, three Indonesian Telco's firms (Telkom, Indosat, and XL) provided the model with data to experiment it. Furthermore, the results disclose the IT value engineering model has worked, and where the model has provided the firms with the improved performance at lower costs. In other words, this model has been effective numerically as well as logically.

Efficient XVA Computation under Local LLvy Models

Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FBSDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local Levy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.

Unwinding External Stock Imbalances? The Case of Italy’s Net International Investment Position

This paper is a case study of an (almost complete) adjustment of Italy’s external stock imbalance. After reaching a peak of around 25 per cent of GDP in early 2014, Italy’s net external debtor position has steadily decreased, reaching less than 7 per cent of GDP at the end of 2017. The contribution of this work is twofold. First, it reviews the main developments in Italy’s net international investment position (NIIP) since 1999. Second, it reports a baseline projection of Italy’s NIIP over a medium-term horizon, as implied by current account balance forecasts. Since this projection ignores the role of valuation adjustments, the study also provides an analysis of their sensitivity to a set of potential movements in exchange rates and equity or bond markets.