An Accurate Solution for Credit Valuation Adjustment (CVA) and Wrong Way Risk

This paper presents a Least Square Monte Carlo approach for accurately calculating credit value adjustment (CVA). In contrast to previous studies, the model relies on the probability distribution of a

An Accurate Solution for Credit Valuation Adjustment (CVA) and Wrong Way Risk

Tim Xiao

Journal of Fixed Income, 25(1) 84-95 Summer 2015

ABSTRACT

This paper presents a Least Square Monte Carlo approach for accurately calculating credit value adjustment (CVA). In contrast to previous studies, the model relies on the probability distribution of a default time/jump rather than the default time itself, as the default time is usually inaccessible. As such, the model can achieve a high order of accuracy with a relatively easy implementation. We find that the valuation of a defaultable derivative is normally determined via backward induction when their payoffs could be positive or negative. Moreover, the model can naturally capture wrong or right way risk.

Key Words: credit value adjustment (CVA), wrong way risk, right way risk, credit risk modeling, least square Monte Carlo, default time approach (DTA), default probability approach (DPA), collateralization, margin and netting.

For years, a widespread practice in the industry has been to mark derivative portfolios to market without taking counterparty risk into account. All cash flows are discounted using the LIBOR curve. But the real parties, in many cases, happen to be of lower credit quality than the hypothetical LIBOR party and have a chance of default.

As a consequence, the International Accounting Standard (IAS) 39 requires banks to provide a fair-value adjustment due to counterparty risk. Although credit value adjustment (CVA) became mandatory in 2000, it received a little attention until the recent financial crises in which the profit and loss (P&L) swings due to CVA changes were measured in billons of dollars. Interest in CVA began to grow. Now CVA has become the first line of defense and the central part of counterparty risk management.

CVA not only allows institutions to move beyond the traditional control mindset of credit risk limits and to quantify counterparty risk as a single measurable P&L number, but also offers an opportunity for banks to dynamically manage, price and hedge counterparty risk. The benefits of CVA are widely acknowledged. Many banks have set up internal credit risk trading desks to manage counterparty risk on derivatives.

The earlier works on CVA are mainly focused on unilateral CVA that assumes that only one counterparty is defaultable and the other one is default-free. The unilateral treatment neglects the fact that both counterparties may default, i.e., counterparty risk can be bilateral. A trend that has become increasingly relevant and popular has been to consider the bilateral nature of counterparty credit risk. Although most institutions view bilateral considerations as important in order to agree on new transactions, Hull and White (2013) argue that bilateral CVA is more controversial than unilateral CVA as the possibility that a dealer might default is in theory a benefit to the dealer.

CVA, by definition, is the difference between the risk-free portfolio value and the true (or risky or defaultable) portfolio value that takes into account the possibility of a counterparty’s default. The risk-free portfolio value is what brokers quote or what trading systems or models normally report. The risky portfolio value, however, is a relatively less explored and less transparent area, which is the main challenge and core theme for CVA. In other words, central to CVA is risky valuation.

In general, risky valuation can be classified into two categories: the default time approach (DTA) and the default probability approach (DPA). The DTA involves the default time explicitly. Most CVA models in the literature (Brigo and Capponi (2008), Lipton and Sepp (2009), Pykhtin and Zhu (2006) and Gregory (2009), etc.) are based on this approach.

Although the DTA is very intuitive, it has the disadvantage that it explicitly involves the default time. We are very unlikely to have complete information about a firm’s default point, which is often inaccessible (see Duffie and Huang (1996), Jarrow and Protter (2004), etc.). Usually, valuation under the DTA is performed via Monte Carlo simulation. On the other hand, however, the DPA relies on the probability distribution of the default time rather than the default time itself. Sometimes the DPA yields simple closed form solutions.

The current popular CVA methodology (Pykhtin and Zhu (2006) and Gregory (2009), etc.) is first derived using DTA and then discretized over a time grid in order to yield a feasible solution. The discretization, however, is inaccurate. In fact, this model has never been rigorously proved. Since CVA is used for financial accounting and pricing, its accuracy is essential. Moreover, this current model is based on a well-known assumption, in which credit exposure and counterparty’s credit quality are independent. Obviously, it can not capture wrong/right way risk properly.

In this paper, we present a framework for risky valuation and CVA. In contrast to previous studies, the model relies on the DPA rather than the DTA. Our study shows that the pricing process of a defaultable contract normally has a backward recursive nature if its payoff could be positive or negative.

An intuitive way of understanding these backward recursive behaviours is that we can think of that any contingent claim embeds two default options. In other words, when entering an OTC derivatives transaction, one party grants the other party an option to default and, at the same time, also receives an option to default itself. In theory, default may occur at any time. Therefore, the default options are American style options that normally require a backward induction valuation.

Wrong way risk occurs when exposure to a counterparty is adversely correlated with the credit quality of that counterparty, while right way risk occurs when exposure to a counterparty is positively correlated with the credit quality of that counterparty. For example, in wrong way risk exposure tends to increase when counterparty credit quality worsens, while in right way risk exposure tends to decrease when counterparty credit quality declines. Wrong/right way risk, as an additional source of risk, is rightly of concern to banks and regulators. Since this new model allows us to incorporate correlated and potentially simultaneous defaults into risky valuation, it can naturally capture wrong/right way risk.

The rest of this paper is organized as follows: Section 2 discusses unilateral risky valuation and unilateral CVA. Section 2 elaborates bilateral risky valuation and bilateral CVA. Section 3 presents numerical results. The conclusions are given in Section 4. . All proofs and a practical framework that embraces netting agreements, margining agreements and wrong/right way risk are contained in the appendices.

  1. Unilateral Risky Valuation and Unilateral CVA

It is well-known that the survival probability from time t to s in this framework is defined by

The default probability for the period (t, s) in this framework is defined by

where

Next, we turn to risky valuation. In a unilateral credit risk case, we assume that party A is default-free and party B is defaultable. Risky valuation can be generally classified into two categories: the default time approach (DTA) and the default probability (intensity) approach (DPA).

Although the DTA is very intuitive, it has the disadvantage that it explicitly involves the default time/jump. We are very unlikely to have complete information about a firm’s default point, which is often inaccessible. Usually, valuation under the DTA is performed via Monte Carlo simulation.

Similarly, we have

In theory, a default may happen at any time, i.e., a risky contract is continuously defaultable. This Continuous Time Risky Valuation Model is accurate but sometimes complex and expensive. For simplicity, people sometimes prefer the Discrete Time Risky Valuation Model that assumes that a default may only happen at some discrete times. A natural selection is to assume that a default may occur only on the payment dates. Fortunately, the level of accuracy for this discrete approximation is well inside the typical bid-ask spread for most applications (see O’Kane and Turnbull (2003)). From now on, we will focus on the discrete setting only, but many of the points we make are equally applicable to the continuous setting.

Proposition 1: The unilateral risky value of the single-payment contract in a discrete-time setting is given by

where

Proof: See the appendix.

Proposition 2: The unilateral risky value of the multiple-payment contract is given by

Proof: See the appendix.

For an intuitive explanation, we can posit that a defaultable contract under the unilateral credit risk assumption has an embedded default option (see Sorensen and Bollier (1994)). In other words, one party entering a defaultable financial transaction actually grants the other party an option to default. If we assume that a default may occur at any time, the default option is an American style option. American options normally have backward recursive natures and require backward induction valuations.

The similarity between American style financial options and American style default options is that both require a backward recursive valuation procedure. The difference between them is in the optimal strategy. The American financial option seeks an optimal value by comparing the exercise value with the continuation value, whereas the American default option seeks an optimal discount factor based on the option value in time.

The unilateral CVA, by definition, can be expressed as

Proposition 2 provides a general form for pricing a unilateral defaultable contract. Applying it to a particular situation in which we assume that all the payoffs are nonnegative, we derive the following corollary:

Corollary 1: If all the payoffs are nonnegative, the risky value of the multiple-payments contract is given by

The CVA in this case is given by

The current popular CVA model (e.g., equation (17) in Pykhtin and Zhu (2007) and equation (3) in Gregory (2009)) is quite different from above either equation (12) or equation (14). As a matter of fact, the current CVA model has never been rigorously proved. In order to reflect the economic value of counterparty credit risk, to measure the profit and loss of a bank and to provide proper incentives to traders, a good CVA model must be not only rigorous and accurate but also feasible to implement.

  1. Bilateral Risky Valuation and Bilateral CVA

There is ample evidence that corporate defaults are correlated. The default of a firm’s counterparty might affect its own default probability. Thus, default correlation and dependence arise due to the counterparty relations. Default correlation can be positive or negative. The effect of positive correlation is usually called contagion, whereas the latter is referred to as competition effect.

Table 1. Payoffs of a bilaterally defaultable contract

Proposition 3: The bilateral risky value of the single-payment contract is given by

where

Proof: See the appendix.

Proposition 4: The bilateral risky value of the multiple-payment contract is given by

Proposition 4 says that the pricing process of a multiple-payment contract has a backward nature since there is no way of knowing which risk-adjusted discounting rate should be used without knowledge of the future value. Only on the maturity date, the value of the contract and the decision strategy are clear. Therefore, the evaluation must be done in a backward fashion, working from the final payment date towards the present. This type of valuation process is referred to as backward induction.

There is a common misconception in the market. Many people believe that the cash flows of a defaultable financial contract can be priced independently and then be summed up to give the final risky price of the contract. We emphasize here that this conclusion is only true of the financial contracts whose payoffs are always positive. In the cases where the promised payoffs could be positive or negative, the valuation requires not only a backward recursive induction procedure, but also a strategic selection of different discount factors according to the market value in time. This coupled valuation process allows us to capture correlation between counterparties and market factors.

The bilateral CVA of the multiple-payment contract can be expressed as

  1. Numerical Results

In this section, we present some numerical results for CVA calculation based on the theory described above. First, we study the impact of margin agreements on CVA. The testing portfolio consists of a number of interest rate and equity derivatives. The number of simulation scenarios (or paths) is 20,000. The time buckets are set weekly. If the computational requirements exceed the system limit, one can reduce both the number of scenarios and the number of time buckets. The time buckets can be designed fine-granularity at the short end (e.g., daily and then weekly) and coarse-granularity at the far end (e.g. monthly and then yearly). The rationale is that the calculation becomes less accurate due to the accumulated error from simulation discretization, and inherited errors from calibration of the underlying models, such as those due to the change of macro-economic climate. The collateral margin period of risk is assumed to be 14 days (2 weeks).

Table 4. The impact of the both collateral thresholds on the CVA

Next, we examine the impact of wrong way risk. Wrong way risk occurs when exposure to a counterparty is adversely correlated with the credit quality of that counterparty, while right way risk occurs when exposure to a counterparty is positively correlated with the credit quality of that counterparty. Wrong/right way risk, as an additional source of risk, is rightly of concern to banks and regulators.

Some financial markets are closely interlinked, while others are not. For example, CDS price movements have a feedback effect on the equity market, as a trading strategy commonly employed by banks and other market participants consists of selling a CDS on a reference entity and hedging the resulting credit exposure by shorting the stock. On the other hand, Moody’s Investor’s Service (2000) presents statistics that suggest that the correlations between interest rates and CDS spreads are very small.

To capture wrong/right way risk, we need to determine the dependency between counterparties and to correlate the credit spreads or hazard rates with the other market risk factors, e.g. equities, commodities, etc., in the scenario generation.

Table 5. The impact of wrong way risk on the CVA

  1. Conclusion

This article presents a framework for pricing risky contracts and their CVAs. The model relies on the probability distribution of the default jump rather than the default jump itself, because the default jump is normally inaccessible. We find that the valuation of risky assets and their CVAs, in most situations, has a backward recursive nature and requires a backward induction valuation. An intuitive explanation is that two counterparties implicitly sell each other an option to default when entering into an OTC derivative transaction. If we assume that a default may occur at any time, the default options are American style options. If we assume that a default may only happen on the payment dates, the default options are Bermudan style options. Both Bermudan and American options require backward induction valuations.

Based on our theory, we propose a novel cash-flow-based framework (see appendix) for calculating bilateral CVA at the counterparty portfolio level. This framework can easily incorporate various credit mitigation techniques, such as netting agreements and margin agreements, and can capture wrong/right way risk. Numerical results show that these credit mitigation techniques and wrong/right way risk have significant impacts on CVA.

Appendix

  1. Proofs

Proof of Proposition 1: Under the unilateral credit risk assumption, we only consider the default risk when the asset is in the money. Assume that a default may only occur on the payment date. Therefore, the risky value of the asset at t is the discounted expectation of all possible payoffs and is given by

where

where

Similarly, we have

Proof of Proposition 3: We assume that a default may only occur on the payment date. At time T, there are four possible states: 1) both A and B survive, 2) A defaults but B survives, 3) A survives but B defaults, and 4) both A and B default. The joint distributions of A and B are given by (15). Depending on whether the payoff is in the money or out of the money at T, we have

where

  1. A practical framework for calculating bilateral CVA

We develop a practical framework for calculating bilateral CVA at counterparty portfolio level based on the theory described above. The framework incorporates netting and margin agreements, and captures right/wrong way risk.

Two parties are denoted as A and B. All calculations are from the perspective of party A. Let the valuation date be t. The CVA computation procedure consists of the following steps.

B.1. Risk-neutral Monte Carlo scenario generation

One core element of the trading credit risk modeling is the Monte Carlo scenario generation (market evolution). This must be able to run a large number of scenarios for each risk factor with flexibility over parameterization of processes and treatment of correlation between underlying factors. Credit exposure may be calculated under real probability measure, while CVA or pricing counterparty credit risk should be conducted under risk-neutral probability measure.

Due to the extensive computational intensity of pricing counterparty risk, there will inevitably be some compromise of limiting the number of market scenarios (paths) and the number of simulation dates (also called “time buckets” or “time nodes”). The time buckets are normally designed fine-granularity at the short end and coarse-granularity at the far end. The details of scenario generation are beyond the scope of this paper.

B.2. Cash flow generation

For ease of illustration, we choose a vanilla interest rate swap, as interest rate swaps collectively account for around two-thirds of both the notional and market value of all outstanding derivatives (FinPricing (2015))

Terms

Rates

Figure B1: An interest rate swaplet

Cash flow generation for products without early-exercise provision is quite straightforward. For early-exercise products, one can use the approach proposed by Longstaff and Schwartz (2001) to obtain the optimal exercise boundaries and then the payoffs.

B3. Aggregation and netting agreements

After generating cash flows for each deal, we need to aggregate them at counterparty portfolio level at each scenario and each time bucket. The cash flows are aggregated by either netting or nonnetting based on the netting agreements. A netting agreement is a provision that allows the offset of settlement payments and receipts on all contracts between two counterparties. Another important use of netting is the close-out netting that allows the offset of close-out values.

For netting, we add all cash flows together at the same scenario and the same time bucket to recognize offsetting. The aggregated cash flow under netting at scenario j and time bucket k is given by

For nonnetting, we divided cash flows into positive and negative groups and add them separately. In other words, the offsetting is not recognized. The aggregated cash flows under nonnetting at scenario j and time bucket k are given by

B4. Margin (or collateral) agreements

For a more detailed discussion on pricing collateralized contract/portfolio, see Xiao (2013b).

B5. CVA Calculation

After aggregating all cash flows via netting, one can price a portfolio in the same manner as pricing a single deal. We assume that the reader is familiar with the least square Monte Carlo valuation model proposed by Longstaff and Schwartz (2001) and thus do not repeat some well-known procedures for brevity.

If the counterparty portfolio is collateralized, we can calculate the risky value based on equation (21) of Xiao (2013b). If there is no collateral agreement, we can price the portfolio according to Proposition 4 in this paper.

CVA is by definition the difference between the risk-free portfolio value and the true (or risky or defaultable) portfolio value.

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