Credit Spread Options Research Articles

ADI method of credit spread option pricing based on jump-diffusion model

As the main contribution of this article, we establish an option on a credit spread under a stochastic interest rate. The intense volatilities in financial markets cause interest rates to change greatly; thus, we consider a jump term in addition to a diffusion term in our interest rate model. However, this decision leads us to a partial integral differential equation. Since the integral part might bring some difficulties, we put forward a fairly new numerical scheme based on the alternating direction implicit method. In the remainder of the article, we discuss consistency, stability, and convergence of the proposed approach. As the final step, with the help of the MATLAB program, we provide numerical results of implementing our method on the governing equation.

A Stochastic Correlation Model with Time Change for Pricing Credit Spread Options

In this paper, we introduce the stochastic correlation processes for modeling the credit spread. We first model the components of spread process as correlated Ornstein-Uhlenbeck processes and correlation as Jacobi process. Using the properties of Jacobi process, we are able to obtain the analytical solutions for the credit spread option prices. To further enhance the model’s ability to capture the abrupt changes in the observed correlation time series, we construct a new model where the correlation is modeled by a Jacobi process time change by Levy subordinators. We employ the eigenfunction expansion methods to obtain the closed-form solutions for the option prices. Our empirical study indicates the time changed Jacobi process fits the correlation series significantly better than the Jacobi process.

Pricing credit spread option with Longstaff-Schwartz and GARCH models in Chinese bond market

This paper investigates the mean-reversion and volatile of credit spread time series by using regression and time series analysis in Chinese bond market. Then the Longstaff-Schwartz model and GARCH model are applied to price credit spread put option. The authors compare the features of these two models by employing daily bond prices of government bonds and corporate bonds for the period 2010–2012 in Chinese bond market. The proposed results show that the higher the credit ratings of the corporate bonds are, the lower the prices of the credit spread options are.

Split bond ratings and rating migration

This paper examines the relationships between split ratings and ratings migration. We find that bonds with split ratings are more likely to have future rating changes. A one-notch (more-than-one-notch) split rating increases the probability of rating change within one year of initial issuance by about 3% (6%). Furthermore, we find that about 30% of split rated bonds have their two ratings converge after four years of initial issuance. The rating convergence tapers off after three years, and the rating agency with a higher (lower) initial rating generally maintains a higher (lower) rating in subsequent years if the two ratings do not converge. We also show that rating transition estimation can be improved by taking into consideration split ratings. We find that one-year rating transition matrices are significantly different between non-letter-split rated bonds and letter-split rated bonds, and we show that the difference has an economically significant impact on the pricing of credit spread options and VaR-based risk management models. Overall, our results suggest that split ratings contain important information about subsequent rating changes.

Dynamic Term Structure Modeling: The Preface

Dynamic Term Structure Modeling, the second book in the trilogy on the Fixed Income Valuation Course, shows how to value interest rate derivatives and credit derivatives using a variety of affine, quadratic, HJM, and LIBOR market models. Using a new taxonomy, this book classifies all term structure models as either fundamental models, or preference-free single-plus, double-plus, and triple-plus models. Filled with in-depth insights and expert advice, this book shows how to price basic interest rate and credit derivative products, such as Treasury and Eurodollar futures, bond options, interest rate options (e.g., caps, floors, and swaptions), forward rate agreements, interest rate swaps, credit default swaps, credit spread options, and others. With intuitive explanations and fully developed examples, this book provides new transforms for building efficient trees under state-dependent volatility models, stochastic volatility models, and jump-diffusion models, for pricing American options, and describes fast computational methods such as the Fourier inversion method (including the FFT) and the cumulant expansion method, for valuing interest rate derivatives and credit derivatives, under a variety of affine, quadratic, and LIBOR market models. This book is also accompanied by an informative CD-ROM, which contains various Excel®/VBA® spreadsheets. This software allows for valuation of interest rate derivatives by building interest rate trees for low-dimensional affine models, as well as computing solutions using quasi-analytical formulas for higher-dimensional affine, quadratic, and LIBOR market models. Though most of the programs require coding in advanced scientific languages, such as C and C++, the final output is presented in user-friendly Excel/VBA spreadsheets.

Valuing credit derivatives in a jump-diffusion model

This paper presents a simple framework for valuing single-name credit derivatives in jump-diffusion models. The Gaver–Stehfest algorithm is used to calculate the CDS spread when the value of the reference entity is assumed to the double exponential jump-diffusion process. We model directly the credit spread using a geometric Ornstein–Uhlenbeck process with a jump and derive the pricing formula for credit spread option.

On pricing of credit spread options

This paper describes and analyses different pricing models for credit spread options such as Longstaff–Schwartz, Black, Das–Sundaram and Duan (GARCH-based) models. The first two models, Longstaff–Schwartz and Black, assume respectively a mean-reverting dynamic and a lognormal distribution for the spread and are representative of the so-called “spread models”. Such models consider the spread as a unique variable and provide closed form solutions for option pricing. On the contrary Das–Sundaram propose a recursive backward induction procedure to price credit spread options on a bivariate tree, which describes the dynamic of the term structure of forward risk-neutral spread and risk-free rate. This model belongs to the class of structural models, which can be used to price a wider range of credit risk derivatives. Finally, we consider the pricing of credit spread options assuming a discrete time GARCH model for the spread.

Pricing credit spread options under a Markov chain model with stochastic default rate

AbstractThis paper studies a Markov chain model that, unlike the existing models, has a stochastic default rate model so as to reflect real world phenomena. We extend the existing Markov chain models as follows: First, our model includes both the economy‐wide and the rating‐specific factors, which affect credit ratings. Second, our model allows both continuous and discrete movements in credit spreads, even when there exist no changes in credit ratings. Under these assumptions, we provide a valuation formula for a credit spread option, and examine its effects. This paper suggests a parsimonious model. As in J. Wei (2003), we find that rating‐specific factors are important. Also, discrete movements seem to play a larger role depending on the firm's credit rating. Finally, we show that a model, like the Kodera model, that uses only a common factor without allowing for discrete movements, may overprice credit spread put options. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:631–648, 2004

Valuing credit derivatives using Gaussian quadrature: A stochastic volatility framework

AbstractThis article proposes semi‐closed‐form solutions to value derivatives on mean reverting assets. A very general mean reverting process for the state variable and two stochastic volatility processes, the square‐root process and the Ornstein‐Uhlenbeck process, are considered. For both models, semi‐closed‐form solutions for characteristic functions are derived and then inverted using the Gauss‐Laguerre quadrature rule to recover the cumulative probabilities. As benchmarks, European call options are valued within the following frameworks: Black and Scholes (1973) (represents constant volatility and no mean reversion), Longstaff and Schwartz (1995) (represents constant volatility and mean reversion), and Heston (1993) and Zhu (2000) (represent stochastic volatility and no mean reversion). These comparisons show that numerical prices converge rapidly to the exact price. When applied to the general models proposed (represent stochastic volatility and mean reversion), the Gauss‐Laguerre rule proves very efficient and very accurate. As applications, pricing formulas for credit spread options, caps, floors, and swaps are derived. It also is shown that even weak mean reversion can have a major impact on option prices. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:3–35, 2004

On the Pricing of Credit Spread Options: A Two Factor HW–BK Algorithm

In this article we describe what a credit spread option (CSO) is and show a tree algorithm to price it. The tree algorithm we have opted for is a two factor model composed by a Hull and White (HW) one factor for the interest rate process and a Black-Karazinsky (BK) one factor for the default intensity. As opposed to the tree model of Schonbucher 1999 the intensity process cannot become negative. Having as input the risk free yield curve and market implied default probability curve the model by construction will price correctly the associated defaultable bond. We then use Market data to calibrate the model to price an at the money (ATM) CSO call and then test it to price an out of the money (OTM) Bermudan CSO call on a CDS. Furthermore the discussions in this paper show in practice the difficulties and challenges faced by financial institutions in marking to market those instruments.

No-Arbitrage Approach to Pricing Credit Spread Derivatives

Among the new credit-based derivatives are options on the yield spread between a credit-sensitive instrument and a comparable riskless bond. Valuation requires considering the behavior of both the risk-free interest rate and the credit spread, which may be correlated. A challenge in implementing valuation models for such derivatives is the need to match observed market pricing for both riskless and risky securities. In this article, Chu and Kwok show how a pricing model for credit spread options can be constructed and calibrated to the market, while maintaining its ability to produce option prices in closed form. The two correlated processes are assumed to follow the “extended Vasicek” model developed by Hull and White for arbitrage-free modeling of the interest rate. The article also shows how models of the Black-Karasinski and Heath-Jarrow-Morton type may also be fitted, though not in closed form.

A Gaussian Term Structure Model of Credit Spreads and Valuation of Credit Spread Options

A Gaussian Term Structure Model of Credit Spreads and Valuation of Credit Spread Options

Credit risk: The case of First Interstate Bankcorp

Structural models for pricing credit risk can be used to forecast the spread on risky bonds and for hedging credit risk. This article examines the forecasting accuracy of the Black–Scholes–Merton (BSM) model of risky debt using a data set consisting of weekly bond data for First Interstate Bancorp over the period January 1986–August 1993. In addition, structural model hedge parameters and credit spread options are tested for their effectiveness in hedging the increasing credit risk premium on First Interstate Bancorp debt. Credit spread options in combination with a duration hedge offer the best hedging strategy, reducing the standard deviation of the hedging error by a minimum of 84%.

Coping with Credit Risk

Credit risk is pervasive throughout financial markets. Traditionally, various financial institutions have assumed the burden of credit risk. Banks have supported the credit risk attached to bank loans and forward contracts. Credit insurance companies have provided coverage for the commercial credit risk faced by suppliers of consumer and investment goods and services. Public insurers, such as the ECGD in the UK, have specialized in the coverage of credit risk attached to export trade and overseas investment. Specialized institutions, such as factoring companies, have offered credit risk coverage as one component in a basket of financial services. More recently, the proliferation of financial contracts that entail counter-party default risk - such as swaps, back-to-back loans, and derivative products - have focused attention on ways to deal with credit risk in the marketplace. New products, such as credit default swaps, credit spread options and total-rate-of-return swaps, have allowed firms and financial institutions to more effectively deal with credit risks. Indeed, a recent survey by the British Banker's Association estimates the current market for credit derivative to be around $900 billion in notional principal, with that amount expected to jump to over $1.5 trillion in 2002. In addition, insurance markets have reacted with an array of new products, many with the backing of larger capital markets, such as insurance-linked securities and finite-risk contracts (Shimpi 1999).

A Markov Chain Model with Stochastic Default Rate for Valuation of Credit Spreads

“The effort to bring default risk into our pricing paradigm is a major growth area within the field of derivatives. One of the newly standard approaches to this topic uses a Markov transition matrix to model the process of Ocredit migrationO across standard bond rating agency rating classes, and potentially into default. Early efforts along these lines produced unrealistic paths for bond yields, because credit spreads among rating classes were assumed to be fixed. KoderaOs model introduces a time-varying risk of default, which produces much more familiar diffusion-like behavior for credit spreads. A practical application of the model illustrates the importance of spread volatility in pricing credit spread options.”

A correction note on the first passage time of an Ornstein-Uhlenbeck process to a boundary

This paper provides the derivation of the hitting time density of an Ornstein-Uhlenbeck process to a flat boundary. The derivation relies on a change of measure approach and delivers an explicit formula. This formula is an amended expression of the result given in Leblanc and Scaillet (1998). It corresponds to the formula given by a time substitution approach when the boundary level coincides with the mean of the invariant measure. It can for example be used to price digital up-and-in credit spread options when the logarithm of the credit spread is assumed to follow an Ornstein-Uhlenbeck process.

Default risk in a market model

This paper presents a theoretical model of default risk in the context of the “market” model approach to interest rate dynamics. We propose a model for finite-interval interest rates (such as LIBOR) which explicitly takes into account the possibility of default through the influence of a point process with deterministic intensity. We relate the defaultable interest rate to the non-defaultable interest rate and to the credit risk characteristics default intensity and recovery rate. We find that the spread between defaultable and non-defaultable rate depends on the non-defaultable rate even when the default intensity is deterministic. Prices of a cap on the defaultable rate and of a credit spread option are derived. We consider swaps with unilateral and bilateral default risk and derive the fair fixed swap rate in both cases. Under the condition that both counterparties are of the same risk class, we show that for a monotonously increasing term structure the swap rate for a defaultable swap will lie below the swap rate for a swap without default risk.

Modeling Term Structures of Defaultable Bonds

This article presents convenient reduced-form models of the valuation of contingent claims subject to default risk, focusing on applications to the term structure of interest rates for corporate or sovereign bonds. Examples include the valuation of a credit-spread option. This article presents a new approach to modeling term structures of bonds and other contingent claims that are subject to default risk. As in previous “reduced-form” models, we treat default as an unpredictable event governed by a hazard-rate process. 1 Our approach is distinguished by the parameterization of losses at default in terms of the fractional reduction in market value that occurs at default. Specifically, we fix some contingent claim that, in the event of no default, pays X at time T . We take as given an arbitrage-free setting in which all securities are priced in terms of some short-rate process r and equivalent martingale measure Q [see Harrison and Kreps (1979) and Harrison and Pliska (1981)]. Under this “risk-neutral” probability measure, we letht denote the hazard rate for default at time t and let Lt denote the expected fractional loss in market value if default were to occur at time t , conditional