Pricing Of Credit Derivatives Research Articles

Pricing of Credit Risk Derivatives with Stochastic Interest Rate

This paper deals with a credit derivative pricing problem using the martingale approach. We generalize the conventional reduced-form credit risk model for a credit default swap market, assuming that the firms’ default intensities depend on the default states of counterparty firms and that the stochastic interest rate follows a jump-diffusion Cox–Ingersoll–Ross process. First, we derive the joint Laplace transform of the distribution of the vector process (rt,Rt) by applying piecewise deterministic Markov process theory and martingale theory. Then, using the joint Laplace transform, we obtain the explicit pricing of defaultable bonds and a credit default swap. Lastly, numerical examples are presented to illustrate the dynamic relationships between defaultable securities (defaultable bonds, credit default swap) and the maturity date.

Copula sensitivity analysis for portfolio credit derivatives

Modeling dependence among input random variables is often critical for performance evaluation of stochastic systems, and copulas provide one approach to model such dependence. In the financial industry, the dependence among default times of different assets drives the pricing of portfolio credit derivatives, including basket default swaps and collateralized debt obligations. Copula sensitivities provide information about how changes in the dependence level affect the output. We study sensitivity analysis of elliptical copulas and Archimedean copulas using infinitesimal perturbation analysis, and an unbiased estimators is derived using conditional Monte Carlo (CMC) to address discontinuities that arise in portfolio credit derivatives. In simulation experiments, the new estimators have smaller variance than other applicable methods.

Basket Credit Derivative Pricing in a Markov Chain Model with Interacting Intensities

In this paper, we propose a Markov chain model to price basket credit default swap (BCDS) and basket credit-linked note (BCLN) with counterparty and contagion risks. Suppose that the default intensity processes of reference entities and the counterparty are driven by a common external shock as well as defaults of other names in the contracts. The stochastic intensity of the external shock is a Cox process with jumps. We derive recursive formulas for the joint distribution of default times and obtain closed-form premium rates for BCDS and BCLN. Numerical experiments are performed to show how the correlated default risks may affect the premium rates.

From equity to default correlation with taxes

For fixed income investment, the preponderant risk is the clustering of defaults in the portfolio. Accurate prediction of such clustering depends on the knowledge of default correlation. We develop models with exogenous debt and endogenous debt to predict default correlations from equity correlations based on a self-consistent structural framework. We also examine how taxes affect the prediction of default correlations based on the two models. The empirical analysis shows that the corporate taxes tend to decrease default correlations, while personal taxes could increase or decrease default correlations. Our default correlation model with exogenous debt does a better job of predicting default correlations for high quality bonds, while the one with endogenous debt predicts more accurately for lower rated bonds. Our studies not only theoretically improve the modeling of default correlation in the structural setting but also shed new light on various aspects of default correlations and thereby help financial practitioners price credit derivatives more accurately and formulate more effective strategies to manage default risk of credit portfolios.

Market Expectations and Default Risk Premium in Credit Default Swap Prices: A Case Study of Argentine Default

This article explores expectations of credit market and default risk premium implicit in credit default swap prices in the case of Argentine default. We find that default risk premium is substantial and its relation with default probability is not monotone. Default risk premium increases with a rising default probability when default probability is low, but it declines when default probability reaches a certain level and continues to rise. This result suggests that default risk premium plays a primary role in the pricing of credit derivatives when default risk is moderate, but its impact becomes secondary when default risk becomes extremely high. We also find default risk premium has a hump-shaped term structure with the peak of the curve around five to six years. Our results show that variation in default risk premium in Argentine sovereign debt was affected by changes in the U.S. business cycle and credit conditions and the overall strength of the Argentine economy. Comparing with market expected default probabilities, we find that major rating agencies assigned overly optimistic ratings to Argentine debt and they lagged the market in downgrading the debt.

Increment Variance Reduction Techniques with an Application to Multi-name Credit Derivatives

Increment variance reduction techniques are add-ons to Monte Carlo (MC) simulations. They make MC simulations converging faster by repeating the number of simulations with an incremental rate derived from mathematical functions. Besides speeding up MC simulations, the major advantage of increment techniques is their ability to handle large numbers of simulations avoiding memory saturation and overflow which occur when a plain MC simulation is involved in the pricing of multi-name credit derivatives. A trend among authors pricing financial securities with MC simulation has been to choose Quasi-Monte Carlo (QMC) methods using deterministic sequences instead of MC methods involving pseudorandom generators such as congruential generator and Mersenne twister. The Increment family models circumvent the constraint of identifying the optimal QMC sequence to price a given security by using a common generator such as Matlab-LCG-Xor RNG and determining the optimal mathematical function of incrementation of MC simulations that will make the pricing of the security adequate. Market participants in need of selecting a reliable numerical method for pricing complex financial securities such as multi-name credit derivatives will find our paper appealing.

Quantifying Correlation Uncertainty Risk in Credit Derivatives Pricing

We propose a simple but practical methodology for the quantification of correlation risk in the context of credit derivatives pricing and credit valuation adjustment (CVA), where the correlation between rates and credit is often uncertain or unmodelled. We take the rates model to be Hull–White (normal) and the credit model to be Black–Karasinski (lognormal). We summarise recent work furnishing highly accurate analytic pricing formulae for credit default swaps (CDS) including with defaultable Libor flows, extending this to the situation where they are capped and/or floored. We also consider the pricing of contingent CDS with an interest rate swap underlying. We derive therefrom explicit expressions showing how the dependence of model prices on the uncertain parameter(s) can be captured in analytic formulae that are readily amenable to computation without recourse to Monte Carlo or lattice-based computation. In so doing, we crucially take into account the impact on model calibration of the uncertain (or unmodelled) parameters.

Firm-specific credit risk estimation in the presence of regimes and noisy prices

Security prices are important inputs for estimating credit risk. Yet, to obtain an accurate firm-specific credit risk assessment, one needs a reliable model and a methodology that filters the elements unrelated to the firm’s fundamentals from market prices.In this article, we introduce a hybrid credit risk model defined in a Markov-switching environment. It captures firm-specific changes in the leverage uncertainty during crises. We also propose a new efficient method to estimate the model, and a numerical scheme based on trinomial lattices to price credit derivatives. The estimation is finally performed on more than 200 firms using maximum likelihood estimation.

Pricing credit derivatives under fractional stochastic interest rate models with jumps

Based on the reduced-form approach, this paper investigates the pricing problems of default-risk bonds and credit default swaps (CDSs) for a fractional stochastic interest rate model with jump under the framework of primary-secondary. Using properties of the quasi-martingale with respect to the fractional Brownian motion and the jump technique in Park (2008), the authors first derive the explicit pricing formula of defaultable bonds. Then, based on the newly obtained pricing formula of defaultable bonds, the CDS is priced by the arbitrage-free principle. This paper presents an extension of the primary-secondary framework in Jarrow and Yu (2001).

Interacting default intensity with a hidden Markov process

In this paper we consider a reduced-form intensity-based credit risk model with a hidden Markov state process. A filtering method is proposed for extracting the underlying state given the observation processes. The method can be applied to a wide range of problems. Based on this model, we derive the joint distribution of multiple default times without imposing stringent assumptions on the form of default intensities. Closed-form formulas for the distribution of default times are obtained which are then applied to solve a number of practical problems such as hedging and pricing credit derivatives. The method and numerical algorithms presented can be applicable to various forms of default intensities.

Pricing credit derivatives under a correlated regime-switching hazard processes model

In this paper, we study the valuation of a single-name credit default swap and a $k$th-to-default basket swap under a correlated regime-switching hazard processes model. We assume that the defaults of all the names are driven by a Markov chain describing the macro-economic conditions and some shock events modelled by a multivariate regime-switching shot noise process. Based on some expressions for the joint Laplace transform of the regime-switching shot noise processes, we give explicit formulas for the spread of a CDS contract and the $k$th-to-default basket swap.

Pricing, Valuation and Hedging of Credit Derivatives

In this manuscript we reviews theoretical approaches for pricing, valuation and hedging of Credit Derivatives.

Pricing and hedging of derivatives in contagious markets

It is well documented that stock markets are contagious. A negative shock to one market increases the probability of adverse shocks to other markets. We model this contagion effect by including mutually exciting jump processes in the dynamics of the indexes’ log-returns. On top of this we add a stochastic volatility component to the dynamics. It is important to take the contagion effect into account if derivatives written on a basket of assets are to be priced or hedged. Due to the affine model specification the joint characteristic function of the log-returns is known analytically, and for two specifications we detail how the model can be calibrated efficiently to option prices. In total we calibrate over an extended period of time the specifications to options data on four US stock indexes, and show how the models achieve satisfactory pricing errors. We study the effect of contagion on multi-asset derivatives prices and show how for certain derivatives the impact is heavy. Moreover, we derive hedge ratios for European put and call options and perform a numerical experiment, which illustrates the impact of contagious jumps on option prices and hedge ratios. Mutually exciting processes have been analyzed for multivariate intensity modeling for the purpose of credit derivatives pricing, but have not been used for pricing/hedging options on equity indexes.

Analysis of the Contagion Effect to the Credit Derivative Valuation

This study explores a credit derivative pricing model with counterparty risk and the contagion effect. To compare with the standard credit derivative pricing model, we analyze the counterparty risk and the contagion effect to a kth-to-default Basket Credit Linked Note (BCLN) valuation by Monte Carlo simulation. Counterparty risk and the contagion effect show significant influence for kth-to-default BCLN valuation. Especially with high k, the BCLN pricing model with the counterparty risk and the contagion effect captures chain defaulting phenomenon successfully. It indicates the higher the kth-to-default BCLN or the lower the correlation degree within reference entities, the more significant the contagion effect becomes to the kth-to-default BCLN valuation. Parameters sensitivity analyses indicate that the coupon rates of the third-to-default BCLN are lower with the higher risk-free rate or the shorter maturity. The higher hazard rate or the lower recovery rate of reference entities results in the higher coupon rates of the third-to-default BCLN. Lastly, the hazard rate and the recovery rate of the counterparty are less sensitive to the BCLN pricing. The major contribution in this study is that the credit derivative pricing model with the contagion effect is developed to capture the default chain reaction, and the valuation performance is significant numerically.

Modeling default dependence with the mixture of calendar time and business time

Evaluating default correlation between securities in a portfolio is very important for credit derivatives pricing and risk management. Under the framework of the structural model proposed by Black and Cox, we assume that the asset values of companies are driven by Brownian motions in the worlds of the calendar time and the business time; they then could evolve continuously or by leap. We build the dynamic default correlations using the time-varying correlated Brownian motions in these processes. The sensitivity of default correlations to the key parameters is explored in the paper by numerical examples, and we apply the model to risk management as well. Because default times are unpredictable in the proposed model, the defaults might occur suddenly. Independent defaults and complete correlated defaults can be described in the model as well.

Credit Derivatives Pricing Model for Fuzzy Financial Market

With various categories of fuzziness in the market, the factors that influence credit derivatives pricing include not only the characteristic of randomness but also nonrandom fuzziness. Thus, it is necessary to bring fuzziness into the process of credit derivatives pricing. Based on fuzzy process theory, this paper first brings fuzziness into credit derivatives pricing, discusses some pricing formulas of credit derivatives, and puts forward a One-Factor Fuzzy Copula function which builds a foundation for portfolio credit products pricing. Some numerical calculating samples are presented as well.

The Pricing of Credit Derivatives and Estimation of Default Probability

Under the native-born model of default and the circular model of default, we take the price of credit derivatives into account. It’s supposed that the short-term market interest rates are based on Vasicek model in this article. Firstly, we calculate the price of default-free bonds in zero-coupon bond. Then, we give the default-intensity expressions under the two models. We calculate the prices of default-free bonds under the two default models. For different situations, we estimate the parameters by maximum likelihood estimation method and calculate the default probability of the company. From the analysis of the result, we find that the result conforms to reality. So the models of default intensity we suppose in the bond pricing are reasonable.

Default barrier intensity model for credit risk evaluation

We establish the Default Barrier Intensity (DBI) model, based on the conditional survival probability (also called hazard function barrier), which allows the pricing of credit derivatives with stochastic parameters. Moreover, the DBI is an analytic model which combines the structural and the reduced form approaches. It deals with the impact of the default barrier intensity on the processes around the firm. Using this model we prove the Doob–Meyer decomposition of the default process associated with the random barrier. In this framework, we present the default barrier process as the sum of its compensator (which is a predictable process) and a martingale related to the smallest filtration making the random barrier a stopping time. Furthermore, the DBI as well as the Shifted Square Root Diffusion (SSRD) Alfonsi’s model emphasizes on the dependence between the stochastic default intensity and the interest rate. This model can be useful since it can be easily generalized to all the credit derivatives products such as Collateralized Debt Obligations (CDO) and Credit Default Swaps (CDS).

A reduced-form intensity-based model under fuzzy environments

The external shocks and internal contagion are the important sources of default events. However, the external shocks and internal contagion effect on the company is not observed, we cannot get the accurate size of the shocks. The information of investors relative to the default process exhibits a certain fuzziness. Therefore, using randomness and fuzziness to study such problems as derivative pricing or default probability has practical needs. But the idea of fuzzifying credit risk models is little exploited, especially in a reduced-form model. This paper proposes a new default intensity model with fuzziness and presents a fuzzy default probability and default loss rate, and puts them into default debt and credit derivative pricing. Finally, the simulation analysis verifies the rationality of the model. Using fuzzy numbers and random analysis one can consider more uncertain sources in the default process of default and investors’ subjective judgment on the financial markets in a variety of fuzzy reliability so as to broaden the scope of possible credit spreads.

Pricing and Hedging of Derivatives in Contagious Markets

It is well documented that stock markets are contagious. A negative shock to one market increases the probability of adverse shocks to other markets. We model this contagion effect by including mutually exciting jump processes in the dynamics of the index's log-returns, so that a jump in one market increases the intensities of more jumps in the same market and in other markets. Between jumps the intensities revert to their long-run means. On top of this we add a stochastic volatility component to the dynamics. It is important to take the contagion effect into account if derivatives written on a basket of assets are to be priced or hedged. Due to the affine model specification the joint characteristic function of the log-returns is known analytically, and for two specifications we detail how the model can be calibrated efficiently to option prices using Fourier transform methods. In total we calibrate over an extended period of time the specifications to options data on four US stock indexes; the Amex Biotechnology Index, the Morgan Stanley Technology index, the Securities Broker/Dealer index, and the Natural Gas index, and investigate the effect of contagion on multi-asset derivatives prices. Moreover, we compute the model hedge ratios for put and call options and investigate the historical hedging performances of the contagion specifications. Mutually exciting processes have been analyzed for multivariate intensity modeling for the purpose of credit derivatives pricing, but have not been used for pricing/hedging options on equity indexes.