Mathematical Problems in Engineering

Volume 2015, Article ID 959312, 6 pages

http://dx.doi.org/10.1155/2015/959312

## Calculation of Credit Valuation Adjustment Based on Least Square Monte Carlo Methods

^{1}School of Management, Harbin Institute of Technology, Harbin 150001, China^{2}School of Finance, Harbin University of Commerce, Harbin 150028, China

Received 16 June 2014; Accepted 24 November 2014

Academic Editor: Mohamed A. Seddeek

Copyright © 2015 Qian Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Counterparty credit risk has become one of the highest-profile risks facing participants in the financial markets. Despite this, relatively little is known about how counterparty credit risk is actually priced mathematically. We examine this issue using interest rate swaps. This largely traded financial product allows us to well identify the risk profiles of both institutions and their counterparties. Concretely, Hull-White model for rate and mean-reverting model for default intensity have proven to be in correspondence with the reality and to be well suited for financial institutions. Besides, we find that least square Monte Carlo method is quite efficient in the calculation of credit valuation adjustment (CVA, for short) as it avoids the redundant step to generate inner scenarios. As a result, it accelerates the convergence speed of the CVA estimators. In the second part, we propose a new method to calculate bilateral CVA to avoid double counting in the existing bibliographies, where several copula functions are adopted to describe the dependence of two first to default times.

#### 1. Introduction

For nearly three decades, the deepening of financial innovation made the efficiency of social resource allocation gradually increase. The scale of the global financial derivatives market has been far greater than the total size of the global economy. Most global derivatives trading occurs in the OTC market; therefore, counterparty credit risk management for OTC derivatives needs to be focused on. Counterparty credit risk is complex, resulting in the closure of many large financial institutions, thereby causing thousands of counterparties associated with them to suffer huge losses, so how to estimate counterparty credit risk is becoming increasingly important.

Credit crisis and the ongoing European sovereign debt crisis have highlighted the native form of credit risk, namely, the counterparty risk (Crépey et al. [1]). Counterparty credit risk is defined as the risk that the counterparty to a derivative transaction could default before the final settlement of the transaction cash flows (Brigo and Pallavicini [2]). Since the subprime crisis and default of many banks afterwards, counterparty credit risk has been taken into account by most financial institutions since the regulator has required banks to keep additional capital charges to compensate counterparty risk. Zhang and Wang [3] developed fast and accurate numerical solutions by using fast Fourier transform (FFT) technique. The simulations show that the SVDEJD model is suitable for modeling the long-time real-market changes and stock returns are negatively correlated with volatility. The model and the proposed option pricing method are useful for empirical analysis of asset returns and managing credit risk. Li et al. [4] propose an improved attribute bagging method, weight-selected attribute bagging (WSAB), to evaluate credit risk. And the experimental results based on two credit benchmark databases show that the method, WSAB, is outstanding in both prediction accuracy and stability.

Credit valuation adjustment (CVA) is defined as the difference between the risk-free portfolio value and the true portfolio value that takes into account the possibility of institution and counterparty default (Brigo and Pallavicini [5]). We call it unilateral CVA when only the counterparty has the potential to default and the institution is assumed to be risk-free. Likewise, bilateral CVA assumes that both parties have the possibility to default before the maturity. As in most existing bibliographies such as Zhu and Pykhtin [6], Alavian et al. [7], and Brigo and Pallavicini [8], the authors mainly focus on the details of contracts like collateral agreements, rehypothecation, and close-out rules. Wu et al. [9] propose a stochastic DEA model considering undesirable outputs with weak disposability which not only can deal with the existence of random errors in the collected data, but also depicts the production rules uncovered by weak disposability of the undesirable outputs. This model introduces the concept of risk to define the efficiency of decision making units (DMUs) and utilizes the correlationship matrix of all the variables to portray the weak disposability. Albanese et al. [10] introduce an innovative theoretical framework for the valuation and replication of derivative transactions between defaultable entities based on the principle of arbitrage freedom. The framework extends the traditional formulations based on credit and debit valuation adjustments (CVA and DVA for short, resp.). Bo and Capponi [11] obtain an explicit formula for the bilateral counterparty valuation adjustment of a credit default swaps portfolio referencing an asymptotically large number of entities. They have the result that counterparty adjustments are highly sensitive to portfolio credit risk volatility as well as to the intensity of the common jump process. Rohan and Dmitry [12] think that the effect of counterparty credit on valuation and risk management dramatically increased. Existing modeling and infrastructure no longer worked and a rethink from first principles had to take place. So they evolved a new generation of interest rate modeling based on dual curve pricing and integrated credit valuation adjustment (CVA) is evolving. Jimmy et al. [13] analyse the impact of wrong-way risks on both CVA itself and tail risks of CVA. They demonstrate that the induced tail risk adjustment to account for wrong-way risk can be more significant than the impact on CVA itself. We hope this paper would serve as a complement to the existing research, especially in terms of simulation methods.

As universally agreed, least square Monte Carlo method (Longstaff and Schwartz [14]; Thom [15]) is efficient in the valuation of American or Bermudan options. In essence, the valuation of CVA shares the same principle as both require us to calculate conditional expectation as fast as possible. Therefore, this paper is focused on how to apply least square Monte Carlo method to calculate unilateral CVA (UCVA, for short) and then bilateral CVA (BCVA, for short) with the help of copula functions. We find that this method is impeccable and hope it would become a standard in the CVA valuation.

#### 2. Unilateral Credit Valuation Adjustment

In credit risk modeling, usually we define the probability space and the default time of institution and its counterparty by and , respectively. We define the augmented filtration , where is the usual filtration containing all market information except defaults, while carries only the information of defaults. Besides, we define the default intensity and cumulated default intensity as Then, first time to default (FTD) is defined as ( is standard uniform random variable) Moreover, due to the fact that default intensity is always positive, we have Thus, the following lemma about conditional expectation will help us calculate CVA: The proof is given in Lee et al. [16].

After modeling default events, we need to explore on the other side all the cash flows during the contract’s life. For simplicity, we assume that collateralization or funding is not involved; thus mark-to-market (MtM, for short) value of the contract is defined as the conditional expectation of all the discounted cash-flow with respect to the filtration .

Normally, in the event where a counterparty has defaulted, an institution may close out the position and is not obliged to make future contractual payments (reasonably, since payments are unlikely to be received). However, the underlying contracts must be settled depending on the MtM value at the time of default. Consider the impact of positive or negative MtM with a counterparty in default.(i)*Positive MtM.* When a counterparty defaults, they will be unable to make future commitments and hence an institution will have a claim on the positive MtM at the time of the default. The amount of this MtM minus any recovery value will represent the loss due to the default. (ii)*Negative MtM.* In this case, an institution owes its counterparty through negative MtM and is still legally obliged to settle this amount. Hence, from a valuation perspective, the position is essentially unchanged. An institution does not gain or lose from their counterparty’s default in this case.Therefore, when we calculate UCVA, we only take the positive part of MtM into consideration. If we consider further the recovery rate and the discount factor denoted by , we deduce the following formula on the assumption of no wrong-way risk or, more specifically, no dependence between recovery rate, default, and exposure.

A concrete example would be swap, whose fixed/floating leg value is the conditional expectation of all the coupons paid in the future discounted to the current time. Namely, cash-flow is the sum of all the coupons paid at each future instant. Consider According to the formula of UCVA, we need to take the expectation twice so that a natural idea comes to mind, to say we generate random variables to calculate the inner expectation and to calculate the outer expectation. Graphically we have Figure 1.