More details on the origin of the cash flows used here are in Pallavicini et al. Thus, the risk-neutral price of cash flows due to financing positions recorded in timetjis.

## 3 Generalized Derivatives Valuation

### Discrete-Time Solution

Recall the financing account definitions in (9) if collateral re-hypothecation is not permitted and in (10) if re-hypothecation is-. The ± sign in the theorem is intended to emphasize the fact that the sign of the financing account, which determines the effective financing rate, depends on the sign of the conditional expectation.

### Continuous-Time Solution

The first term is the value of the transaction cash flow, discounted at the funding rate plus credit. The second term is the price of the non-default cash flow in excess of the collateral, which includes the CVA and DVA of the transaction after collateralization.

## 4 Nonlinear Valuation: A Numerical Analysis

*Monte Carlo Pricing**Case Outline**Preliminary Valuation Under Symmetric Funding and Without Credit Risk**Complete Valuation Under Credit Risk, Collateral, and Asymmetric Funding**Nonlinearity Valuation Adjustment*

Finally, if the trader decides to hedge only the risk-free price of the trade, i.e. we assume that the margin account is equal to the risk-free price of the trade on each margin date, i.e. Ct = Vt.

## 5 Conclusions and Financial Implications

As a result, the value of the trade will generally be different from the two counterparties. A trade should be priced at an appropriate level of aggregation to quantify the value it actually adds to the business.

Pallavicini, A., Brigo, D.: Modeling Interest Rates in Collateralized Markets: Multiple Curves, Credit Liquidity Effects, CCPs.arXiv ssrn.com (2013). Pallavicini, A., Perini, D., Brigo, D.: Financing, insurance and hedging: uncovering the mechanics and subtleties of financing valuation adjustments.

## Under Credit and Funding Effects

Nonlinear Valuation of Insurance, Credit Risk and Funding Costs: A Numerical Case Study Extending Black–Scholes, [5]).

## 1 Introduction

Section 2 introduces the probabilistic setting, cash flow analysis, and derives the first valuation equation based on conditional expectations. Section 3 derives the FBSDE under no-default filtration from the initial valuation equation under the assumptions of conditional independence of default times and initial cash flows of the no-default portfolio.

## 2 Cash Flows Analysis and First Valuation Equation

### The Cash Flows

The payments due to the collateral account: More precisely, we model this account according to a predictable process. Since the collateral taker must reimburse the account at the rate, the discounted flows due to the collateral can be expressed as transportation costs.

### Adjusted Cash Flows Under a Simple Trading Model

I return the money I just acquired to the treasury, so the net value of the repo was. I have to return the collateral with interest, so I request the amount of Ct(1+ctdt) from the Treasury, which I return to the counterparty.

## 3 An FBSDE Under F

0 λXsdswithX∈ {I,C}we have that τX =Λ−X1(ξX)with ξX mutually independent exponential random variables independent of λX.1A similar result will enable us to deal with the default cash flow term. We now postulate a special form for the default cash flow, more precisely if we show the process fitted to VttheF such that.

## 4 Markovian FBSDE and PDE for V t and the Invariance Theorem

We now show that a solution to Eq. 13) can be obtained by the classical solution of PDE (17). Properly accounting for the funding costs of a position leads to the funding valuation (AUV) adjustment.

## 2 Prices 2.1 Setup

### Clean Price

Consequently, at time τ (if

All-Inclusive Price

## 3 TVA BSDEs

*Full TVA BSDE**Partially Reduced TVA BSDE**Fully Reduced TVA BSDE**Marked Default Time Setup*

Furthermore, under mild assumptions (see e.g. Crépey and Song [6, Theorem 4.1]), one can easily check that f¯t(ϑ) in (7) (resp.ft(ϑ)) corresponds to the classical BSDE montonicity assumption is satisfied. and likewise forf), for some constantC. In fact, by Lemma 5.1 in Crépey and Song [6], there exist G-predictable processes PteandΔet such that.

## 4 TVA Numerical Schemes 4.1 Linear Approximation

### Linear Expansion and Interacting Particle Implementation

Collecting the terms of the same order with respect to εin (16), we obtain Θt(0)=0, due to the zero-terminal state of the fully reduced BSDE (III), and . The manageability of the FT schemes (23) and (24) depends on the nullity of the terminal state of the related BSDEs (III) and (II), implying that Θ¯(0)=Θ(0)=0.

### Marked Branching Diffusion Approach

If1τ

## 5 TVA Models for Credit Derivatives

### Dynamic Gaussian Copula TVA Model

*Model of Default Times**TVA Model*

181], in the case of vanilla credit derivatives on the reference names, namely CDS contracts and CDO tranches (cf. 47)), there exists a continuous, explicit function, i.e. Assume that the processes r andλ¯ are given beforeτ as continuous functions of (t,Xt), which also applies toPin in the case of vanilla credit derivatives on names in N.

### Dynamic Marshall–Olkin Copula TVA Model

*Model of Default Times**TVA Model*

We assume that the processes r and λ¯ are given in advanceτ as continuous functions of (t, Xt), which also applies to Pin the case of vanilla credit derivatives in names in N. Then the coefficientsf¯ and in the curve are given deterministically in terms of belonging factor processes as. so we are in the Markovian configuration where the FT and PHL schemes are valid and, in principle, viable. A DMO configuration can be used as a TVA model for credit derivatives, with .. 205], in the case of CDS contracts and CDO tranches, for each shock ∈Y and processU=PorΔ, there exists a continuous, clear function UY such .

Strong Versus Weak Dynamic Copula Model

## 6 Numerics

### Numerical Results in the DGC Model

4 LeftTVA in a CDS calculated by the FT scheme of order 3 as a function of the DGC correlation parameter ρ. Fairly similar results with respect to a portfolio of CDS contracts with ten different names. In Fig.5, the left graph shows that the errors, in terms of relative standard errors (% rel. SE), of the different rankings of the FT scheme do not explode with the dimension (the number of credit names underlying the CDS contracts ).

### Numerical Results in the DMO Model

The FT scheme based on the partially reduced TVA BSDE (II) gives an efficient way to estimate the TVA. The counterparty (or the bank) is taken as the eleventh (or tenth) safest name in the portfolio.

## 7 Conclusion

Brigo, D., Chourdakis, K.: Counterparty risk for credit default swaps: the impact of spread volatility and default correlation. Brigo, D., Capponi, A., Pallavicini, A.: Valuation of counterparty risk without arbitrage under collateralization and application to credit default swaps.

However, Hull and White rely on a clear choice of default model and a clear union. A brief introduction to bilateral counterparty risk is given in Section 2a, before the decomposition of BCVA into its building blocks in Section 3 is carried out.

## 2 Counterparty Default Risk

Considering the default risk of both parties, the amount of bilateral CVA is summarized in the following well-known theorem, which basically goes back to Sorensen and Bollier [19]. Based on Theorem 1, the general approach to calculating the counterparty risk-adjusted value, VAD(t,T) is to first determine the risk-free value VA(t,T) of the transaction.

## 3 The Main Building Blocks of CVA

Thus, it may be necessary to discretize the state space of the subtracted exposure residual process. ThenX is replaced by a representative value for this component (usually an average value) for each of the components, and the probabilities of the discretized process are set according to the original probabilities of each component (see the default bucket approach).

## 4 Models for Counterparty Risk

### Independence of CVA Components

This is true as long as the discounted present value without default risk is independent of the credit quality of each counterparty. A major advantage of our approach is that an arbitrary credit risk model can be used instead, since ultimately only the distribution of the default indicator δ matters.

### Modeling Options on the Basis Transaction

However, CVA calculations typically require a trade-off between the accuracy of the model and the efficiency of the calculations. It should be noted that since the financial market usually provides adequate prices of liquid derivatives, any reasonable model can be calibrated to these market prices. Therefore, in the following we can assume that the market-implied distribution of the discounted exposure process is fully known and available.

### Hybrid Models—An Example

For this reason, CVA calculations typically use simpler models than other pricing applications.

## 5 Tight Bounds on CVA

### Tight Bounds on CVA by Mass Transportation

With some abuse of notation, the measure PDenotes the joint distribution of the default process and the exposure processX. Obviously, the smallest and largest CVA that can be obtained from any P that is consistent with the given marginals is given by .

An Alternative Formulation as Assignment Problem

## 6 Example 6.1 Setup

*Counterparty’s Default Modeling**Counterparty Exposure Modeling**Results**Computation Time, Choice of Algorithm, and Impact of Assumptions*

Analogous to fig. 1, we have for each of the four cases a separate subplot, and the left plots again belong to cases 1 and 2. Theoretically, the calculation of the limits boils down to the solution of a linear programming problem.

## 7 Conclusion and Outlook

Brigo, D., Morini, M., Pallavicini, A.: Counterparty credit risk, collateral and financing: with pricing examples for all asset classes. Cespedes, JCG, Herrero, JA, Rosen, D., Saunders, D.: Effective modeling of wrong-way risk, counterparty credit risk capital, and alpha in Basel II.

## CVA with Wrong-Way Risk in the Presence of Early Exercise

We only need to consider the randomness incorporated in the counterparty default probabilities by means of the stochastic hazard rate and price CVA with standard techniques. The paper is organized as follows: in Section 2, we review the Hull–White model for CVA in the presence of WWR and the recursive approach in [1].

## 2 CVA Pricing and WWR

Obtaining this utility function is the most difficult part of the calibration procedure because it involves a "delicate" path-dependent problem that is difficult to implement for realistic portfolios. The process introduced in (6) can be seen as the driver of the stochasticity in survival probabilities, and it plays a key role in bypassing path dependence in the calibration of a(t), as shown in the following proposition.

## 3 The Impact of Early Exercise

### The Pricing Problem

However, consistent with actual practice in the CVA calculation, we assume that counterparty default plays no role in defining the dealer's exercise strategy. 5If we describe the dynamics of the price of a company share, we assume – for the sake of clarity – that such an entity is not subject to default risk.

### The Plain Vanilla Case

1 CVA profiles for European and American options as a function of WWR parameterb for various levels of cost of carry (CoC). 3 Difference in CVA between European and American options as a function of WWR parameter band validity.

## 4 The Bermudan Swaption Case

The upper chart reports the case with no initial lockout period, while in the lower chart we assume that the option cannot be exercised for the first two years. In this case, the moneyness WWR effect is reversed: the more money the option yields, the more relevant the WWR effect becomes.

## 5 Concluding Remarks

Brigo, D., Morini, M., Pallavicini, M.: Counterparty Credit Risk Collateral and Funding: With Pricing Cases for All Asset Classes. Li, M., Mercurio, F.: Jumping with default: wrong-way-risk modeling for credit valuation adjustment.http://ssrn.com/abstract.

## Simultaneous Hedging of Regulatory and Accounting CVA

The other term, σhed2 , can be interpreted as the capital demand for the market risk of the hedging instruments. The hedge amount that minimizes the hedge results in the optimal allocation between CVA risk mitigation and income statement volatility.

## 2 Counterparty Risk from a Regulatory Perspective

In the following, we will considerσsyna's function of the hedge amount and search for its minimum. For technical reasons, we exclude index hedges in the derivation of the optimal hedging strategy.

## The Standardized CVA Risk Charge

### Standardized CVA Risk Charge as Volatility

In this section we will show that the regulators' modeling assumptions behind the standardized CVA risk charge are given by normally distributed CVA returns which are aggregated using a one-factor Gaussian copula model.5 We consider counterparties. The above lemma shows that the standardized CVA risk charge is essentially the sum volatility.

## 3 Counterparty Risk from an Accounting Perspective

### CVA Hedging from an Accounting Perspective

To be more precise, the CDS default should offset CVA moves. Thus, CVA hedging risk differs whether considered from an accounting or regulatory perspective.

## 4 Portfolio P&L

Portfolio P&L Without CVA

Impact with CVA

### Impact of CVA Risk Charge Hedging on the Accounting P&L Volatility

*Definition of the Steering Variable*

The cross terms (third, fourth and fifth sums) describe the interactions between the volatility of the hedging instruments, the CVA and the residual positions. The control variable is given by the synthetic volatility, consisting of the sum of the regulatory CVA volatility and the accounting profit and loss volatility caused by the hedging instruments:.

## 5 Determination of the Optimal Hedge Strategy

### Special Cases

So if we ignore the discrepancy between the accounting and legal CVA, the optimal hedge solution is only provided by the optimal hedge solution of the CVA risk burden alone. This is the case if the risk factors of the remaining positions are highly correlated with the risk factor of the hedging instrument.

Capital Optimization Through an Innovative CVA Hedge

## 1 Preface

The analysis leads to a definition of the concept of liquidity and its relation to the use of collateral in financial markets. Furthermore, with regard to the associated valuation and risk, the liquidity transformation shows similarities with the concept of wrongful risk.

## 2 The Role of Collateral in OTC Contracts and Its Legal Basis

*The Role of Legal Versus Economic Ownership**Affected Market Participants**Financial Instruments Involving Collateral and Standard Legal Frameworks (Master Agreements)**Derivatives Under ISDA Master Agreement**Repos Under GMRA**Securities Lending Under GSLMA**Credit and Counterparty Risk Related to Collateral*

The type and use of collateral are regulated in the CSA (credit support annex), which is an integral part of the ISDA Master Agreement framework1 and cannot be considered separately. The CSA defines the type(s) of collateral and the terms of coverage/disclosure, while the transfer of legal ownership is governed by the ISDA Master Agreement.

## 3 Terms of Liquidity and Definition of Liquidity Transformation

### Terms of Liquidity

If a long position in the underlying securities (collateral) is assumed for Bank 1, the wrong risk involves a decrease in value of the securities (collateral) and a simultaneous decrease in the creditworthiness of Bank 2. In this case, the risk is for Bank 1 is the failure of Bank 2 in balancing the collateral position.

### Comparison of Secured and Unsecured Financing

Secured: Only a residual claim that forms part of the estate of the insolvent party, but the amount of the residual claim is determined independently of the estate of the insolvent party. With secured funding, the default risk is coupled with the recovery risk (price risk) of the collateral and the risk position can be quickly unwound in the event of default, while with unsecured funding, the recovery reversal depends on insolvency. process.

### Liquidity Transformation

This comparison shows in particular that the credit risk towards the counterparty in the unsecured financing transaction which is rather illiquid is opposed to the market value risk of the collateral received which is assumed to be liquid in the secured case plus the correlation of this risk and the credit risk of the issuer of the securities taken as collateral. A bank has different access and a higher degree of freedom to allocate liquidity, regardless of the purpose, than e.g.

## 4 New Approach to CVA Hedging

*Issue**Solution**Application**Example*

With regard to counterparty credit risk, a bank is faced with conflicting objectives as a result of legal requirements, i.e. if the market value of Bank A's derivative trades against counterparty B increases, Bank A is exposed to counterparty credit risk (CVA risk).

## 5 Conclusion

When describing the corresponding cash flow profiles, the two situations, default and non-default of the counterparty, are distinguished (third line in the table above). In the event of counterparty B's default, the residual claim of the transaction is physically delivered to Investment Fund C in exchange for cash equal to the principal amount of the residual claim.

## FVA and Electricity Bill Valuation Adjustment—Much of a Difference?

The following (mildly edited) record of the panel discussion reiterates the main arguments of the debaters - ultimately culminating in the awareness that if everyone charges a value adjustment to the electricity bill, it must become part of any quoted price.

## 1 Welcome

Daniel not only represents one of the main sponsors of this conference, but also represents almost 20 years of experience in the field of financial consulting. Daniel is a member of the financial risk management team at KPMG and has been the responsible partner for risk methodology for over ten years.

## 2 Damiano Brigo

This is a simple accounting rule, but it translates into a rather nightmarish, non-linear constraint on valuation. Whether it is there or not depends on the financing model you have for your treasury.

## 3 Christian Fries

We don't even know exactly which payout we are trading unless we have a very detailed description of the CVA calculation. I don't care if we call the OIS rate the risk-free rate or not, but it's the best near-risk-free benchmark we have.

## 4 John Hull

It's part of the cost of doing business, you're not taking advantage of those 80 basis points. We all know what you're trying to do is get the best price you can and hopefully cover your costs.

## 5 Daniel Sommer

### Acknowledgements, Credits, and Disclaimer

Photos of Damiano Brigo and John Hull by Astrid Eckert; photo of the panel by Bettina Haas. Open Access This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link to the Creative Commons license is provided and any changes made are noted.